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A162169 Signed version of Pascal's triangle. 2

%I

%S 1,-1,1,-1,0,1,1,0,-3,1,1,0,-6,0,1,-1,0,10,0,-5,1,-1,0,15,0,-15,0,1,1,

%T 0,-21,0,35,0,-7,1,1,0,-28,0,70,0,-28,0,1,-1,0,36,0,-126,0,84,0,-9,1,

%U -1,0,45,0,-210,0,210,0,-45,0,1,1,0,-55,0,330,0,-462,0,165,0,-11,1

%N Signed version of Pascal's triangle.

%C Related to A000111 via its matrix inverse A162170.

%C For odd columns k, T(n,k) = binomial(n-1, k-1) * (-1)^floor((n+k-1)/2). For even columns, T(n, k) = 1 if n = k, otherwise 0. - _Mike Tryczak_, Jun 17 2015

%e Table begins:

%e 1;

%e -1, 1;

%e -1, 0, 1;

%e 1, 0, -3, 1;

%e 1, 0, -6, 0, 1;

%e -1, 0, 10, 0, -5, 1;

%e -1, 0, 15, 0, -15, 0, 1;

%t nn=12; Flatten[Table[Table[If[Or[Mod[n - k, 4] == 1, Mod[n - k, 4] == 2], -1, 1]*If[n >= k, Binomial[n - 1, k - 1], 0]*If[And[n > k, Mod[k, 2] == 0], 0, 1], {k, 1, n}], {n, 1, nn}]] (* _Mats Granvik_, Nov 25 2017 *)

%o (Excel) =if(or(mod(row()-column();4)=1;mod(row()-column();4)=2);-1;1)*if(row()>=column();combin(row()-1;column()-1);0)*if(and(row()>column();mod(column();2)=0);0;1)

%o (PARI) T(n, k) = if (k % 2, binomial(n-1, k-1) * (-1)^floor((n+k-1)/2), if (n==k, 1 , 0));

%o tabl(nn) = {for (n=1, nn, for (k=1, n, print1(T(n,k), ", ");); print(););} \\ _Michel Marcus_, Jun 17 2015

%Y Cf. A007318, A162170.

%K sign,tabl

%O 1,9

%A _Mats Granvik_, Jun 27 2009

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Last modified August 8 23:02 EDT 2020. Contains 336300 sequences. (Running on oeis4.)